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4
votes
1answer
185 views

A proof of Theorem 9.2.12. in the Gompf-Stipsicz

I'm seeking for a proof of Theorem 9.2.12. in the Gompf-Stipsicz "4-Manifolds and Kirby Calculus" (for the statement, see the following image). But the textbook omits any proofs and only gives a ...
3
votes
2answers
246 views

CW 4 manifolds with single 4 cell

Let $M$ be a connected compact closed 4 manifold. Then $H_4(M)=\mathbb{Z}$. If we assume it is smooth, from Morse theory we know that $M$ has a CW structure. But can we find a CW structure of $M$ with ...
4
votes
1answer
170 views

How to understand Taubes' moduli space of holomorphic curves?

Let $(X, \omega)$ be a closed symplectic 4-manifold. Let $\mathcal{C}=(C_i, mi_i)$ be a holomorphic current in $X$, where $C_i$ is a somewhere injective $J$-holomorphic curve in $X$ and $m_i$ is ...
3
votes
0answers
72 views

Perburb the Monodromy of Lefschetz fibration over a disk

Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk,(more general, Lefschetz fibration over a surface with boundary ) and let $\Omega$ be a closed 2-form on $E$ such that it ...
0
votes
2answers
129 views

obtaining circle bundle over torus by trefoil surgery

Does any integer surgery on a right or left trefoil knot give the $S^1$-bundle over $T^2$ with Euler number $1$?
6
votes
1answer
304 views

Construction of invariants of 4-manifolds with the Kirby calculus

I'm an undergraduate student, interested in the low dimensional topology, in particular, the 4-manifold theory. I have a question. In the knot theory, the Reidemeister moves play fundamental roles. ...
4
votes
0answers
196 views

Are all 4-manifolds $Pin^{\tilde{c}}$?

It's known that all oriented 4-manifolds admit a $Spin^c$ structure, ie. a spin structure on $TX\oplus\mathcal{L}$ for some complex line bundle $\mathcal{L}$. A usual generalization of this ...
4
votes
1answer
167 views

contact surgery diagram on Brieskorn manifolds

For the Brieskorn manifold $\Sigma(p,q,r)=\{z_1^p+z_2^q+z_3^r=0\} \cap S^5 \subset C^3$, replacing zero with $\epsilon$ in the above, realizes $\Sigma$ as boundary of a Stein domain which induces a ...
4
votes
1answer
159 views

Homology 3-sphere with a unique Stein-fillable contact structure

Are there any known examples of oriented integer homology 3-spheres $Y$ (besides $S^3$) which have exactly one Stein-fillable contact structure up to isotopy? Failing that, what are the known examples ...
8
votes
1answer
180 views

Essential Klein bottle in simply connected symplectic 4 manifolds

Consider the following question: Let $X$ be a simply connected, symplectic 4-manifold. Does there exists a smoothly embedded Klein bottle $K\subset X$ such that the following conditions are both ...
4
votes
0answers
131 views

No irreducible parallelizable manifold of given dimension

What is an example of a closed 4-manifold $M$ such that $M$ is parallelizable and $M$ is topologically (or at least smoothly) irreducible? Topological irreducible: it is not homemorphic to ...
18
votes
0answers
592 views

Concordance and homology cobordism

If two knots $K_1$ and $K_2$ in $S^3$ are smoothly concordant, then for any rational number $r$, the $r$-surgeries $S^3_r(K_1)$ and $S^3_r(K_2)$ are homology cobordant. Is the converse true? What if $...
12
votes
0answers
388 views

Is Rasmussen's s-invariant of a knot an invariant of a 4-manifold?

Let $K$ be a knot in the 3-sphere $S^3$. Here we denote by $s(K)$ Rasmussen's s-invariant for $K$, and by $X_{K}(n)$ the 4-manifold obtained from the standard 4-ball $B^4$ by attaching a $2$-...
3
votes
1answer
177 views

Non-degenerate periodic orbits in the boundary of Lefschetz fibration over a disk

Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk, and let $\Omega$ be a closed 2-form on $E$ such that it is non-degenerate fiberwise. For any $x \in E$, there is a ...
3
votes
2answers
258 views

Legendrian knot in 3-sphere

We are given a Legendrian knot, fixed up to Legendrian isotopy, in $(S^3,\xi)$ ($\xi$ is the standard contact structure). Does it necessarily bound a symplectic surface in $(B^4,\omega)$ (again $\...
3
votes
2answers
218 views

Brieskorn homology spheres

We know that a Brieskorn homology 3-spheres $\Sigma(p,q,r)$ admit a free $S^1$-action, which makes it a Seifert fibered spaces with three singular fibers: $M(b;r_1,r_2,r_3)$. How should one get from $\...
5
votes
1answer
181 views

Embedded spheres in the K3 surface

Using the Seiberg-Witten theory, we know that every (smoothly) embedded $S^{2}$ in $K3$ with trivial normal bundle is null-homologous. We know we have a lot of interesting knotted $S^{2}$ inside $S^{4}...
9
votes
2answers
323 views

Obtain 4-manifolds by repeating surgeries of submanifolds in $S^4$

In his paper QFT and Jones Polynomials, Witten states: "It is a not too deep result that every 3-manifold can be obtained from or reduced to $S^3$ (or any other desired 3-manifold) by repeated ...
5
votes
2answers
246 views

Disjoint curves in an algebraic surface

Let $X$ be an algebraic surface (over the complex) with $p_g=q=0$. Is it possible to have disjoint curves $C_1,\ldots, C_b$, of positive genus, spanning $H_2(X,{\mathbb Q})$, $b=b_2(X)$? (When $X$ is ...
10
votes
1answer
318 views

Piecewise linear (PL) structures on $\mathbf R^4$

One can read in Wikipedia that the 4-dimensional affine space $\mathbf R^4$ has uncountably many piecewise linear structures (in contrast with other dimensions, where it has exactly one). A reference ...
4
votes
0answers
125 views

Blow-up of $\mathbb{P}^4$ along a smooth surface

Let $\pi \colon X\to \mathbb{P}^4$ be the blow-up of a smooth surface $S\subset \mathbb{P}^4$. Is there a formula to compute $(K_X)^4$ ? (which should be dependent on invariants of $S$). In dimension ...
4
votes
1answer
230 views

Are “Unions” of small exotic $\mathbb{R}^4$'s small?

Suppose $M$ is a smooth 4-manifold, and $U,V \subset M$ are exotic $\mathbb{R}^4$'s, i.e. homeomorphic to standard $\mathbb{R} ^4$. Further more suppose $U$ an $V$ intersect nicely sucht that $U \cup ...
15
votes
1answer
591 views

Smooth 4-manifolds with $E_8$ intersection form

Does there exist a closed orientable smooth 4-manifold $M$ whose intersection form is the $E_8$-form? Here by the intersection form I mean the $\mathbb{Z}$-valued bilinear form on $H^2(M;\mathbb{Z})/\...
0
votes
1answer
158 views

relations between intersection form and Chern classes

Is there exist a 4-manifold which intersection form has the following property $$ (a,a) \neq 0\ \text{if}\ a\neq 0, $$ and the second (or the first) Chern class (for some almost complex stucture) ...
2
votes
0answers
155 views

Is complex surface in CP(3) a two handlebody?

Consider a complex surface given by homogeneous equation in $\mathbb{C}P^3$. Without loss of generality, take \begin{equation} S = \{[x:y:z:w] \in \mathbb{C}P^3~ |~ x^d + y^d + z^d + w^d = 0\} \end{...
11
votes
1answer
744 views

Thurston geometries in dimension 4

In the sense of W. Thurston here, there is 3 geometries in dimension 2 and there is 8 geometries in dimension 3. Question: How many different geometries (in the sense of Thurston) do we have in ...
10
votes
0answers
159 views

Excluding exotic PL structures on S^4

Suppose you have a finite group $G<SO(5)$ such that $S^4/G$ is homeomorphic to $S^4$ and such that $S^4/G$ is a PL manifold with respect to a PL structure induced by a standard structure on $S^4$. ...
3
votes
1answer
182 views

Simply-connected 4-manifolds can be blown up and down to complex projective planes. How about non-simply-connected ones?

There is a nice theorem that for every simply connected, closed, smooth, oriented manifold $M$, we have for some $m \in \mathbb{N}$: $$M \# \left(\mathop{\#}^m \left(\mathbb{C}\mathbb{P}^2 \# \...
2
votes
0answers
154 views

Definition of the dual spider number and the formula for the first chern class of the triangle

In the process of trying to understand various maps in Heegaard Floer homology I got stuck on the definition of the dual spider number, which, it seems to me, has a combinatorial definition directly ...
2
votes
2answers
404 views

Approximation theorem for Anti-Self-Dual Metrics

Rounge's Theorem states that any meromorphic function on a domain inside $\mathbb{C}$ can be approximated (over compact subsets) by a sequence of rational functions (meromorphic functions on $\mathbb{...
2
votes
0answers
322 views

Closed 4-manifolds with uncountably many differentiable structures

I know that $\mathbb{R}^4$ admits uncountably many differentiable structures and I was wandering what happen if we consider closed 4-manifolds. Are there any closed 4-manifolds with uncountably many ...
5
votes
1answer
259 views

How to embed genus 4 surface inside $\mathbb{C}P^2\# \mathbb{C}P^2$ representing nontrivial homology class

As the title says, I want to embed the genus 4 surface inside $\mathbb{C}P^2\# \mathbb{C}P^2$ representing a nontrivial homology class. I know that $H_2(\mathbb{C}P^2 \# \mathbb{C}P^2; \mathbb{Z})\...
7
votes
1answer
295 views

Are 4-dimensional mapping tori always spin?

We know that all compact orientable manifolds of dimension 3 are spin. In 4 dimensions, $CP^2$ is not spin. I would like to ask if all 4-dimensional compact orientable mapping tori are spin? See ...
4
votes
1answer
143 views

0-homologous surface bounds

Given a map $f : S \to M^4$ from a compact closed not necessarily connected oriented surface to a compact oriented 4-manifold, such that $f_*([S])$ is zero in $H_2(M)$, is there a compact oriented 3-...
11
votes
0answers
231 views

Fox re-imbedding theorem in dimension four

Fox re-imbedding theorem states the following: A compact 3-manifold $M$ with boundary that embeds in the three-sphere $S^3$, can be re-imbedded in $S^3$ so that its complement is a union of ...
5
votes
1answer
388 views

What are Kirby diagrams of candidate exotic 4-manifolds?

It is an open problem whether there exist smooth manifolds homeomorphic, but not diffeomorphic to the standard $S^4$. The same is true for the 4-torus and several other manifolds. Handle ...
5
votes
2answers
393 views

Does the 4-sphere have a nonzero Poisson structure as a Poisson homogeneous space?

It is known that the 4-sphere does not have a symplectic structure. However, it does admit Poisson structures, for example the zero Poisson structure, which is quite boring. Does it have other, more ...
3
votes
0answers
198 views

Compact self-dual Einstein metrics of negative scalar curvature

Does a compact four-dimensional self-dual Einstein manifold with negative scalar curvature have negative sectional curvature? This would be true if we believe the folklore conjecture that a compact ...
6
votes
2answers
281 views

Knotted projective planes and fake complex projective space

Paul Melvin gave a talk at Knots in Washington last year in which he asked whether the connected sum of an odd twist-spin of a classical knot and a standard cross-cap embedding of ${\mathbb R}P^2$ is ...
0
votes
0answers
110 views

K3 surface minus finite set

Let $S$ be a complex K3 surface, and $P\subset S$ a finite set of points in $S$. It is known that $$ H^i(S,\mathbb{Z})\cong H^i(S\setminus P,\mathbb{Z}) $$ for $0\le i \le 2$. Then the Euler ...
4
votes
1answer
286 views

Casson invariant and signature

In W. Neumann, J. Wahl, "Casson invariant of links of singularities", Comment. Math. Helv.,1990, Vol. 65, Issue 1, pp 58-78 some connection between the Casson invariant and the signature is ...
2
votes
0answers
164 views

complex Morse function on a four-manifold

If we have a complex Morse function on a complex four-manifold, $f: X\to \mathbb{C} $, can we tell from the function how the genus of inverse images $f^{-1}(z)$ (for regular values) may change? under ...
0
votes
1answer
708 views

Stein manifolds definiton

There are a few equivalent definitions of Stein manifolds. As far as I know they were initially defined as holomorphically convex complex manifolds, and then the other definitions (e.g. complex ...
0
votes
0answers
146 views

Affine varieties as Stein surfaces

I have a somewhat general and vague question in mind. Is there anything in literature related to Affine varieties as examples of Stein manifolds? I know that there is a topological approach to Stein ...
2
votes
1answer
271 views

Gluck twist on four-manifolds

I have a basic question which I am not able to figure out. If we do a Gluck twist on a nullhomologous 2-sphere in a 4-manifold, it is said that it does not change its intersection form. But as far as ...
3
votes
1answer
324 views

Boundaries of smooth manifolds

If one has a smooth simply connected manifold $M^n$ which we know to bound an $n+1$ dimensional manifold $N$, what can be said about a handle decomposition for one in terms of a handle decomposition ...
14
votes
3answers
1k views

open problems in Seiberg-Witten Theory on 4-Manifolds

What are some of the open problems in Seiberg-Witten Theory on 4-Manifolds.I tried googling but couldn't any. I tried googling it, but couldn't find any resources.The places where I can a survey or ...
6
votes
0answers
195 views

What is the state of the art in 4-manfold 2-types?

In an old answer to an old question of mine, Peter Teichner commented that it is an open problem to determine which homotopy 2-types arise from 4-manifolds. In some instances we know that a 4-manifold ...
7
votes
1answer
404 views

Almost parallelizable 4-manifolds

On the first page of Milnor-Kervaire's paper "Bernoulli numbers, homotopy groups, and a theorem of Rohlin", they assert without proof or reference that if $M$ is a compact connected oriented ...
10
votes
1answer
275 views

Visualising locally flat embeddings of surfaces in R^4

As far as I understand it follows from the work of M. Freedman that there exist locally flat embeddings of two dimensional surfaces in $\mathbb R^4$ that can not be smoothed in the class of locally ...